##
**An introduction to infinite dimensional dynamical systems – geometric theory. With an appendix by Krzysztof P. Rybakowski.**
*(English)*
Zbl 0533.58001

Applied Mathematical Sciences, Vol. 47. New York etc.: Springer-Verlag. 195 p., 17 figs. DM 54.00; $ 20.20 (1984).

The many applications which involve dynamical systems in infinite dimensional spaces, as dynamical systems generated by partial differential equations and delay differential or functional equations, enforce the efforts for the development of a theory of such systems. This book is dedicated to the study of the dynamical systems in infinite dimensions, the theory of which is analogous to the theory in finite dimension. More precisely, the discussion centers around retarded functional differential equations (RFDE) and a geometric theory for these systems is obtained, although the authors point out when the techniques and results are applicable to a more general abstract framework.

An RFDE \(F\) on a separable \(C^{\infty}\) finite dimensional connected manifold \(M\) is a continuous function \(F: C^0(I,M)\to TM\) such that \(\tau_MF=\rho\), where \(I\) is the closed interval \([-r,0]\), \(r\geq 0\), \(TM\) is the tangent bundle of \(M\), \(\tau_M: TM\to M\) is its \(C^{\infty}\) canonical projection and \(\rho: C^0(I,M)\to M\) is the evaluation map. A solution of an RFDE \(F\) on \(M\) with initial condition \(\phi \in C^0(I,M)\) at \(t_0\) is a continuous function \(x(t)\) with values in \(M\) and defined on \(t_0-r\leq t<t_0+A\) for some \(0<A<\infty\), such that: \(x_{t_0}=\phi\), \(x(t)\) is a \(C^1\)-function of \(t\in [t_0,t_0+A)\) and \((x(t),ẋ(t))=F(x_t)=(x(t),f(x_t))\) for an appropriate function \(f\).

First, an existence and uniqueness theorem for initial value problems is established if RFDE is locally Lipschitzian with basis on the corresponding result for \(M={\mathbb{R}}^n\). Related to this result, the solution map or semiflow of an RFDE\((F)\), \(\Phi(t,\phi,F)=x t(\phi,F)\), is defined and some important properties of the semiflow \(\Phi\) are given.

In continuation, some remarkable examples of RFDE on manifolds are given. Denoting by \({\mathfrak X}^k(I,M)\), \(k\geq 1\), the Banach space of all \(C^k\) RFDE’s defined on the manifold \(M\) which are bounded and have bounded derivatives up to order \(k\), taken with the \(C^k\)-uniform norm, the first generic results for RFDEs were established for equations defined on a compact manifold \(M\), proving that the sets \(G^k_0\) and \(G^k_1\) of all RFDE’s in the Banach space \({\mathfrak X}^ k(I,M)\) which have all critical points nondegenerate and hyperbolic, respectively, are open and dense in \({\mathfrak X}^k(I,M)\) and the sets \(G^k\!_{3/2}(T)\) and \(G^k\!_2(T)\) of all RFDE’s in \({\mathfrak X}^k(I,M)\) for which all nonconstant periodic solutions with period in \((0,T)\) are nondegenerate and hyperbolic, respectively, are open in \(\mathfrak X^k(I,M)\), \(k\geq 1\). For the case of RFDE on \(\mathbb{R}^n\) it is proved that the set of all RFDE’s in \(\mathfrak X^k(I,\mathbb{R}^n)\), \(k\geq 2\), with \(C^k\)-uniform topology which have all critical points and all periodic orbits hyperbolic is residual in \(\mathfrak X^k(I,\mathbb{R}^n)\).

The invariant sets and limit sets of an RFDE\((F)\) are naturally defined and the important properties of these sets are established. Now, given an RFDE\((F)\) on \(M\) and denoting by \(A(F)\) the invariant set of all initial data of global bounded solutions of \(F\), the authors show that a very simple condition of stability at \(t=+\infty\), namely point dissipativeness, implies \(A(F)\) compact and, in this case, it is the maximal compact invariant set of \(F\), it is connected, uniformly asymptotically stable, attracts all bounded sets of \(C^0\) and \(A(F)=\cap_{n\geq 0}\Phi_{nr}K\) where \(K\) is any compact set which attracts all compact sets of \(C^0\). Due to the above properties of the set \(A(F)\) it is naturally called the attractor set of \(F\). Certain continuity properties of the attractor set \(A(F)\) in relation to the dependence on \(F\) and some results on the ”size” of \(A(F)\), \(F\in \mathfrak X^k(I,M)\), \(k\geq 1\), in terms of limit capacity and Hausdorff dimension are given.

Since the study of qualitative properties of the flow is facilitated if \(A(F)\) has a simple geometric structure, the results are established in the direction to determine when the attractor \(A(F)\) is a \(C^1\)-manifold through the use of \(C^k\)-retractions. In fact, if \(F\in \mathfrak X^1(I,M)\), \(M\) is a compact and connected manifold without boundary and there exists a \(C^1\)-retraction \(\gamma: C^0\to C^0\) such that \(A(F)=\gamma(C^0)\) then \(A(F)\) is a connected compact \(C^1\)-manifold. A very restrictive condition for the existence of a such retract is given in Theorem 7.3, namely, the existence of a constant \(k>0\) such that \(\| d\Phi_ t(\phi)\| \leq k\) and \(d\Phi_ t\) has Lipschitz constant \(k\) for all \(t\geq 0\) and \(\phi \in C^0\). However, using infinite dimensional analogues on the continuity properties of a certain class of attractors, the authors show that the attractor set of small perturbations of equations satisfying the above condition are also \(C^1\)-manifolds.

Another class of RFDE’s whose attractors are \(C^1\)-manifolds are the RFDEs close to ordinary differential equations: ”Let \(X\) be a \(C^1\)-vector field defined on a compact manifold \(M\). There is a neighborhood \(v\) of \(F=X\mathbb{O}\rho\) in \(\mathfrak X^1(I,M)\) such that \(A(G)\) is a \(C^1\)-manifold diffeomorphic to \(M\) for \(G\in V\), \(A(G)\to A(F)\) as \(G\to F\) and the restriction of \(\Phi^G\!_t\), \(t\geq 0\), to \(A(G)\) is a one-parameter family of diffeomorphisms” (Theorem 7.6). Also, in the case \(M=\mathbb{R}^n\) and \(F\) is given by an ordinary differential equation a result somewhat similar to Theorem 7.6 is proved using the different considerations.

The primary objective in the qualitative theory of RFDEs, namely the study of the dependence of the flow \(\Phi_t=\Phi^F\!_t\) on \(F\), requires naturally the existence of a criterion for deciding when two RFDEs are equivalent. The authors point out in Section 8 that a study of the dependence of the flow on changes of the RFDE through the use of a notion of equivalence based on a comparison of all orbits is very difficult, the difficulty being associated with the infinite dimensionality of the phase space and the associated smoothing properties of the solution operator. Consequently the authors consider a notion of equivalence which ignores some of the orbits of the RFDEs to be compared; namely, two RFDEs \(F\) and \(G\) in \(\mathfrak X^1(I,M)\) are said to be equivalent, \(F\sim G\), if there is a homeomorphism \(h: A(F)\to A(G)\) which preserves orbits and sense of direction in time. An RFDE \(F\) defined on a manifold is called A-stable if there is a neighborhood \(V\) of \(F\) such that \(G\sim F\) if \(G\in V\). Some special examples which indicate on one hand that the comparison of the flows of \(A(F)\) and \(A(G)\) near an equilibrum point will involve global properties of the flow and, on the other hand, how the set \(A(F)\) may vary with \(F\), are presented. Also, a reasonable formulation of an analogue of the Hartman-Grobman theorem is given.

In Section 9 of the book the authors present a study on equations obtained by compactification of linear delay equations \(\dot x(t)=Ax(t-1)\) in \(\mathbb{R}^2\) and in \(\mathbb{R}\) (compactified to the sphere \(S^2\) and the circle \(S^1\), respectively) in order to illustrate that the idea of compactification of the Euclidean plane into the unit two-dimensional sphere \(S^2\) used by Poincaré for the study of the behavior at infinity of solutions of ordinary differential equations can be applied to RFDEs.

In the last section of the book the authors deal with the maps \(f\) which belong to \(C^r(B,E)\), the Banach space of all \(E\)-valued \(C^r\)-maps defined on \(B\) which are bounded together with their derivatives up to order \(r\geq 1\), \(B\) being a Banach manifold imbedded in a Banach space \(E\). By \(C^r(B,B)\) is denoted the subspace of \(C^r(B,E)\) of all maps leaving \(B\) invariant, by \(A(f)\) the set \(\{x\in B\): there exists a sequence \((x=x_1,x_2,\ldots)\in B\), \(\sup_{j}\| x_j\|<\infty\) and \(f(x_j)=x_{j-1}\), \(j=2,3,\ldots\}\), and by \(KC^r(B,B)\) is denoted the subspace of \(C^r(B,B)\) of all \(f\in C^r(B,B)\), \(f\) reversible (that is \(f/A(f)\) and \(df/A(df)\) are injective maps) has \(A(f)\) compact and given a neighborhood \(U\) of \(A(f)\) in \(B\) there exists a neighborhood \(W(f)\) of \(f\) in \(C^r(B,B)\) such that \(A(g)\subset U\) for all \(g\in W(f)\). A map \(f\in KC^r(B,B)\) is called A-stable if there exists a neighborhood \(V(f)\) of \(f\) in \(KC^r(B,B)\) such that to each \(g\in V(f)\) corresponds a homeomorphism \(h\), \(h=h(g): A(f)\to A(g)\) and \(h.f=g.h\) hold on \(A(f)\). The special set \(MS\) of the Morse-Smale maps of \(KC^r(B,B)\) is considered and studied. So, it is proved that from the dynamics point of view a Morse-Smale map \(f\) exhibits the simplest orbit structure.

Moreover, the following main result of this section is proved: ”The set \(MS\) is open in \(KC^r(B,B)\) and any \(f\in MS\) is A-stable”.

The book finishes with bibliographical notes, references and with an appendix ”An introduction to the homotopy index theory in noncompact spaces” by K. P. Rybakowski.

An RFDE \(F\) on a separable \(C^{\infty}\) finite dimensional connected manifold \(M\) is a continuous function \(F: C^0(I,M)\to TM\) such that \(\tau_MF=\rho\), where \(I\) is the closed interval \([-r,0]\), \(r\geq 0\), \(TM\) is the tangent bundle of \(M\), \(\tau_M: TM\to M\) is its \(C^{\infty}\) canonical projection and \(\rho: C^0(I,M)\to M\) is the evaluation map. A solution of an RFDE \(F\) on \(M\) with initial condition \(\phi \in C^0(I,M)\) at \(t_0\) is a continuous function \(x(t)\) with values in \(M\) and defined on \(t_0-r\leq t<t_0+A\) for some \(0<A<\infty\), such that: \(x_{t_0}=\phi\), \(x(t)\) is a \(C^1\)-function of \(t\in [t_0,t_0+A)\) and \((x(t),ẋ(t))=F(x_t)=(x(t),f(x_t))\) for an appropriate function \(f\).

First, an existence and uniqueness theorem for initial value problems is established if RFDE is locally Lipschitzian with basis on the corresponding result for \(M={\mathbb{R}}^n\). Related to this result, the solution map or semiflow of an RFDE\((F)\), \(\Phi(t,\phi,F)=x t(\phi,F)\), is defined and some important properties of the semiflow \(\Phi\) are given.

In continuation, some remarkable examples of RFDE on manifolds are given. Denoting by \({\mathfrak X}^k(I,M)\), \(k\geq 1\), the Banach space of all \(C^k\) RFDE’s defined on the manifold \(M\) which are bounded and have bounded derivatives up to order \(k\), taken with the \(C^k\)-uniform norm, the first generic results for RFDEs were established for equations defined on a compact manifold \(M\), proving that the sets \(G^k_0\) and \(G^k_1\) of all RFDE’s in the Banach space \({\mathfrak X}^ k(I,M)\) which have all critical points nondegenerate and hyperbolic, respectively, are open and dense in \({\mathfrak X}^k(I,M)\) and the sets \(G^k\!_{3/2}(T)\) and \(G^k\!_2(T)\) of all RFDE’s in \({\mathfrak X}^k(I,M)\) for which all nonconstant periodic solutions with period in \((0,T)\) are nondegenerate and hyperbolic, respectively, are open in \(\mathfrak X^k(I,M)\), \(k\geq 1\). For the case of RFDE on \(\mathbb{R}^n\) it is proved that the set of all RFDE’s in \(\mathfrak X^k(I,\mathbb{R}^n)\), \(k\geq 2\), with \(C^k\)-uniform topology which have all critical points and all periodic orbits hyperbolic is residual in \(\mathfrak X^k(I,\mathbb{R}^n)\).

The invariant sets and limit sets of an RFDE\((F)\) are naturally defined and the important properties of these sets are established. Now, given an RFDE\((F)\) on \(M\) and denoting by \(A(F)\) the invariant set of all initial data of global bounded solutions of \(F\), the authors show that a very simple condition of stability at \(t=+\infty\), namely point dissipativeness, implies \(A(F)\) compact and, in this case, it is the maximal compact invariant set of \(F\), it is connected, uniformly asymptotically stable, attracts all bounded sets of \(C^0\) and \(A(F)=\cap_{n\geq 0}\Phi_{nr}K\) where \(K\) is any compact set which attracts all compact sets of \(C^0\). Due to the above properties of the set \(A(F)\) it is naturally called the attractor set of \(F\). Certain continuity properties of the attractor set \(A(F)\) in relation to the dependence on \(F\) and some results on the ”size” of \(A(F)\), \(F\in \mathfrak X^k(I,M)\), \(k\geq 1\), in terms of limit capacity and Hausdorff dimension are given.

Since the study of qualitative properties of the flow is facilitated if \(A(F)\) has a simple geometric structure, the results are established in the direction to determine when the attractor \(A(F)\) is a \(C^1\)-manifold through the use of \(C^k\)-retractions. In fact, if \(F\in \mathfrak X^1(I,M)\), \(M\) is a compact and connected manifold without boundary and there exists a \(C^1\)-retraction \(\gamma: C^0\to C^0\) such that \(A(F)=\gamma(C^0)\) then \(A(F)\) is a connected compact \(C^1\)-manifold. A very restrictive condition for the existence of a such retract is given in Theorem 7.3, namely, the existence of a constant \(k>0\) such that \(\| d\Phi_ t(\phi)\| \leq k\) and \(d\Phi_ t\) has Lipschitz constant \(k\) for all \(t\geq 0\) and \(\phi \in C^0\). However, using infinite dimensional analogues on the continuity properties of a certain class of attractors, the authors show that the attractor set of small perturbations of equations satisfying the above condition are also \(C^1\)-manifolds.

Another class of RFDE’s whose attractors are \(C^1\)-manifolds are the RFDEs close to ordinary differential equations: ”Let \(X\) be a \(C^1\)-vector field defined on a compact manifold \(M\). There is a neighborhood \(v\) of \(F=X\mathbb{O}\rho\) in \(\mathfrak X^1(I,M)\) such that \(A(G)\) is a \(C^1\)-manifold diffeomorphic to \(M\) for \(G\in V\), \(A(G)\to A(F)\) as \(G\to F\) and the restriction of \(\Phi^G\!_t\), \(t\geq 0\), to \(A(G)\) is a one-parameter family of diffeomorphisms” (Theorem 7.6). Also, in the case \(M=\mathbb{R}^n\) and \(F\) is given by an ordinary differential equation a result somewhat similar to Theorem 7.6 is proved using the different considerations.

The primary objective in the qualitative theory of RFDEs, namely the study of the dependence of the flow \(\Phi_t=\Phi^F\!_t\) on \(F\), requires naturally the existence of a criterion for deciding when two RFDEs are equivalent. The authors point out in Section 8 that a study of the dependence of the flow on changes of the RFDE through the use of a notion of equivalence based on a comparison of all orbits is very difficult, the difficulty being associated with the infinite dimensionality of the phase space and the associated smoothing properties of the solution operator. Consequently the authors consider a notion of equivalence which ignores some of the orbits of the RFDEs to be compared; namely, two RFDEs \(F\) and \(G\) in \(\mathfrak X^1(I,M)\) are said to be equivalent, \(F\sim G\), if there is a homeomorphism \(h: A(F)\to A(G)\) which preserves orbits and sense of direction in time. An RFDE \(F\) defined on a manifold is called A-stable if there is a neighborhood \(V\) of \(F\) such that \(G\sim F\) if \(G\in V\). Some special examples which indicate on one hand that the comparison of the flows of \(A(F)\) and \(A(G)\) near an equilibrum point will involve global properties of the flow and, on the other hand, how the set \(A(F)\) may vary with \(F\), are presented. Also, a reasonable formulation of an analogue of the Hartman-Grobman theorem is given.

In Section 9 of the book the authors present a study on equations obtained by compactification of linear delay equations \(\dot x(t)=Ax(t-1)\) in \(\mathbb{R}^2\) and in \(\mathbb{R}\) (compactified to the sphere \(S^2\) and the circle \(S^1\), respectively) in order to illustrate that the idea of compactification of the Euclidean plane into the unit two-dimensional sphere \(S^2\) used by Poincaré for the study of the behavior at infinity of solutions of ordinary differential equations can be applied to RFDEs.

In the last section of the book the authors deal with the maps \(f\) which belong to \(C^r(B,E)\), the Banach space of all \(E\)-valued \(C^r\)-maps defined on \(B\) which are bounded together with their derivatives up to order \(r\geq 1\), \(B\) being a Banach manifold imbedded in a Banach space \(E\). By \(C^r(B,B)\) is denoted the subspace of \(C^r(B,E)\) of all maps leaving \(B\) invariant, by \(A(f)\) the set \(\{x\in B\): there exists a sequence \((x=x_1,x_2,\ldots)\in B\), \(\sup_{j}\| x_j\|<\infty\) and \(f(x_j)=x_{j-1}\), \(j=2,3,\ldots\}\), and by \(KC^r(B,B)\) is denoted the subspace of \(C^r(B,B)\) of all \(f\in C^r(B,B)\), \(f\) reversible (that is \(f/A(f)\) and \(df/A(df)\) are injective maps) has \(A(f)\) compact and given a neighborhood \(U\) of \(A(f)\) in \(B\) there exists a neighborhood \(W(f)\) of \(f\) in \(C^r(B,B)\) such that \(A(g)\subset U\) for all \(g\in W(f)\). A map \(f\in KC^r(B,B)\) is called A-stable if there exists a neighborhood \(V(f)\) of \(f\) in \(KC^r(B,B)\) such that to each \(g\in V(f)\) corresponds a homeomorphism \(h\), \(h=h(g): A(f)\to A(g)\) and \(h.f=g.h\) hold on \(A(f)\). The special set \(MS\) of the Morse-Smale maps of \(KC^r(B,B)\) is considered and studied. So, it is proved that from the dynamics point of view a Morse-Smale map \(f\) exhibits the simplest orbit structure.

Moreover, the following main result of this section is proved: ”The set \(MS\) is open in \(KC^r(B,B)\) and any \(f\in MS\) is A-stable”.

The book finishes with bibliographical notes, references and with an appendix ”An introduction to the homotopy index theory in noncompact spaces” by K. P. Rybakowski.

Reviewer: Liliana Maxim-Răileanu (Iaşi)

### MSC:

58-02 | Research exposition (monographs, survey articles) pertaining to global analysis |

37Cxx | Smooth dynamical systems: general theory |

58B20 | Riemannian, Finsler and other geometric structures on infinite-dimensional manifolds |